Introduction

Voting and Collective Decision-Making

What Problem Does Voting Solve?

Whenever a group must make a decision -- and its members disagree -- a procedure is required.

• A city must choose a mayor.
• A legislature must fill a vacant seat.
• A community must select a board member.
• A nation must choose a president.

In each case, many individuals have preferences. But only one outcome can occur.

The central problem is this:

How can a group translate many individual preferences into one collective decision?

Voting is one family of solutions to that problem.

It is not the only one. Consensus, delegation, appointment, and random selection have all been used in different contexts. But voting is among the most widely adopted mechanisms for resolving disagreement in modern democracies.

Understanding voting systems requires understanding what they are designed to do.

From Assemblies to Elections

In small groups, decision-making can be direct.

In ancient assemblies, citizens gathered in one place and voted by show of hands or voice. When the group is small and visible, counting can be immediate. Debate and voting occur in the same physical space.

But scale changes design.

As populations grow:

• Gathering everyone in one place becomes impractical.
• Votes must be recorded rather than observed.
• Ballots must be counted reliably.
• Results must be accepted as legitimate.

Over time, representative systems emerged. Instead of voting on every issue directly, citizens elect individuals to make decisions on their behalf.

This shift -- from direct decision to selecting decision-makers -- creates a new design challenge:

How should we choose among multiple candidates competing for a single office?

The answer is not self-evident. Different societies have adopted different rules.

Those rules shape outcomes.

What Voting Systems Actually Do

At a basic level, a voting system performs three functions:

1. Collect information from voters (a single choice, a ranking, approvals, ratings)
2. Process that information according to a counting rule (tallying, elimination rounds, pairwise comparison)
3. Produce a winner

Each of these stages involves design decisions.

• Should voters choose only one candidate?
• Should they rank all candidates?
• Should they indicate which are acceptable?
• Should the winner require a majority?
• Should counting occur in one round or multiple?
• Should the winner be the candidate with the most support? The broadest support? The least opposition?

There is no procedure that optimizes for every possible value at once.

Some systems prioritize simplicity. Some prioritize majority support. Some prioritize expressive ballots. Some attempt to reduce vote splitting.

Every system balances tradeoffs.

This series examines those tradeoffs.

Along the way, we will encounter three recurring concepts that are worth naming up front.

Strategic voting occurs when a voter casts a ballot that does not reflect their genuine preferences because they believe a different ballot will produce a better outcome for them. This is rational behavior within a system's constraints -- not cheating and not a failure of the voter. Different systems create different strategic pressures, and some make honest expression a stronger strategy than others.

Pathologies are systematic outcomes that no reasonable design would intentionally produce -- results that undermine the purpose of holding an election. Vote splitting, where similar candidates cannibalize each other's support and a less-preferred candidate wins as a result, is one example. Pathologies are distinct from tradeoffs, which are consequences that follow from a deliberate design prioritization. A tradeoff may be acceptable depending on what one values; a pathology is not.

Honesty incentives describe the degree to which a voting system rewards voters for expressing their genuine preferences. In some systems, honest expression is frequently the best strategy. In others, voters face strong pressure to misrepresent their preferences in order to achieve a better outcome. How much a system rewards honesty is one of the most practical questions a voter or reformer can ask.

Each of these concepts will be examined in detail as they arise in specific voting systems.

Three Contexts for Voting

Voting is used in several distinct contexts:

1. Single-winner elections -- One seat is filled. (Mayor, Governor, President.)
2. Multi-winner elections -- Multiple seats are filled at once. (City councils, legislatures, boards.)
3. Referendums and ballot measures -- Voters choose among policy options rather than candidates. (Bond measures, ballot propositions.)

Each context presents different design questions.

This series focuses on the first category:

Single-winner elections used to fill a single office.

These elections present a particularly interesting challenge when more than two candidates compete. Questions about majority support, vote splitting, elimination order, and preference expression arise most clearly in this setting.

Multi-winner systems introduce additional structural questions -- such as proportional representation and minority inclusion -- which will be addressed separately.

Referendums operate under a different logic, typically involving binary choices. They raise distinct issues and will not be the focus here.

A Note on Scope

This series does not attempt to promote a particular reform.

Instead, it seeks to build structural literacy.

For each voting system we examine, we will ask:

• What problem is it attempting to solve?
• What information does it collect?
• How does it process that information?
• What tradeoffs does it introduce?
• What outcomes does it guarantee -- and what does it not?

The goal is not to declare one system "best."

It is to understand what each system is designed to prioritize.

Only after understanding those design choices can meaningful evaluation occur.

Where We Begin

We will begin with the most widely used method in the United States and many other countries:

Plurality voting.

Plurality is often treated as the default.

But it, too, is a design choice -- one that emerged under particular historical and administrative conditions.

Understanding why plurality became dominant -- and what it optimizes for -- provides the foundation for evaluating every alternative that follows.

From there, we will examine other single-winner systems that attempt to address its structural consequences.

Each new method will emerge as a response to limitations in the previous one.

And each will introduce tradeoffs of its own.

Plurality Voting

The Most Common Method -- and Its Structural Consequences

Statement of Purpose

This article explains Plurality Voting, the most widely used method for single-winner elections.

We will:

• Describe how plurality ballots are cast and counted
• Examine why the system became dominant
• Identify what it optimizes for
• Explore the structural tradeoffs it introduces

Plurality voting is often treated as the default. But it, too, is a design choice.

Understanding its structure is essential before evaluating alternatives.

Section 1: How Plurality Voting Works

Plurality voting -- also called First Past the Post (FPTP) -- is straightforward:

• Each voter selects one candidate.
• The candidate with the most votes wins.
• A majority is not required.

There are:

• No elimination rounds
• No rankings
• No scoring scales
• No runoffs (unless separately required by law)

Counting consists of a single tally.

A Simple Example

Three candidates:

• Alice 🤟🏼
• Ben 🫰🏼
• Carl ✌🏼

100 voters cast ballots.

Alice wins.

She has more votes than any other candidate.

But she does not have a majority (51 would be required).

Plurality means:

More votes than anyone else -- not more than half.

A Categorical Ballot

The plurality ballot is the simplest type of ballot possible. Voting theorists call it a categorical ballot.

A categorical ballot records a selection -- and nothing else.

It does not capture how voters would rank the candidates. It does not capture how strongly voters feel. It records only one piece of information: which candidate the voter chose.

Later in this series, we will encounter other ballot types:

• Ordinal ballots, which ask voters to rank candidates in order of preference.
• Cardinal ballots, which ask voters to rate candidates on a numerical scale.

Each ballot type captures different information. Each enables different counting methods.

The categorical ballot captures the least information -- but it is also the simplest to cast and count.

Section 2: Why Plurality Became Dominant

Plurality voting did not become widespread by accident.

Its dominance reflects both historical circumstances and practical advantages.

Historical Roots

Modern plurality voting traces largely to the English parliamentary tradition. For centuries, English constituencies elected members by a simple rule: each voter named one candidate, and the candidate with the most support won the seat.

This practice took hold in an era when most contested elections involved only two serious candidates. With two candidates, plurality and majority rule produce the same result -- the candidate with the most votes necessarily has more than half. The structural tensions that arise with three or more candidates were rarely tested.

As English electoral practices spread -- to colonial legislatures, to the early United States, to other parliamentary systems -- plurality came with them. It was embedded in legal frameworks, administrative procedures, and public expectations before multi-candidate elections became routine.

By the time more candidates began competing regularly, plurality was already the institutional default.

Administrative Advantages

The system's persistence also reflects genuine practical strengths:

1️⃣ Simplicity for Voters

• Mark one name.
• No ranking required.
• Minimal ballot instructions.

2️⃣ Simplicity of Counting

• One round.
• No transfers.
• No recalculation.
• Easy to audit and verify.

3️⃣ Speed of Results

Because counting is direct, results can often be determined quickly.

4️⃣ Transparency

The process is intuitive:

• Count the marks.
• Highest total wins.

Plurality scales easily.

It works reliably when:

• Only two candidates compete, or
• One candidate has clear majority support.

In those contexts, plurality produces the same winner as a majority-rule system.

Section 3: Consensus, Majority, and Plurality

When groups make decisions, there are different standards that can determine an outcome.

These standards measure different kinds of agreement.

Agreement Standards Matrix

Consensus

Consensus refers to a condition in which a decision is accepted by all -- or nearly all -- participants.

It is defined primarily by the absence of significant opposition rather than by the presence of enthusiastic support.

A consensus outcome might:

• Not be everyone's first choice
• Not generate strong enthusiasm
• But also not provoke meaningful resistance

Consensus does not have a fixed mathematical threshold. It describes a condition of collective acceptance.

Majority

A majority means:

More than half of the relevant total.

In a 100-voter election, a majority requires at least 51 votes.

Majority rule is an absolute threshold.

A decision can pass with 51% support even if 49% strongly object.

Plurality

Plurality means:

More votes than any other single option.

Plurality is a relative measure:

• The winner has the largest share.
• That share may be well below half.

Plurality does not attempt to achieve consensus or guarantee majority support.

It identifies the largest supported bloc in a single round.

A Three-Candidate Example

• No candidate has consensus.
• No candidate has a majority.
• Alice has a plurality.

Plurality identifies the largest group of supporters.

It does not measure whether most voters accept the outcome.

The Same Election, Viewed Differently

Now look at the same result from the other direction.

Alice won with a plurality. But a majority voted against her. Both statements describe the same election.

Sixty out of 100 voters -- a clear majority -- chose someone other than Alice. Under plurality, those 60 voters are on the losing side despite outnumbering Alice's supporters. Their preferences were divided across two candidates, but their collective opposition to Alice is real and measurable. Plurality has no mechanism to register it.

A plurality yes is a majority no.

This is not a flaw unique to one election or one set of candidates. It is a structural feature of the counting rule. Whenever more than two candidates compete and no one reaches a majority, the plurality winner takes office over the objection of most voters. The system does not ask whether most voters find the outcome acceptable. It asks only who received the largest single share.

Section 4: Vote Splitting

Plurality performs cleanly when the electorate is divided between two clear options.

But when more candidates enter the race, a new dynamic can emerge.

Consider:

A majority (55 voters) prefer a progressive candidate.

But because their support is divided, the conservative candidate wins.

This phenomenon is often called vote splitting or the spoiler effect.

Vote splitting is an example of what we will call a pathology -- a systematic outcome that no reasonable design would intentionally produce. No one designs a voting system hoping that similar candidates will cannibalize each other's support. When vote splitting changes the outcome of an election, the system has produced a result that undermines the purpose of holding the election in the first place.

Pathologies are different from tradeoffs. A tradeoff is a consequence that follows from a deliberate design choice -- it may be acceptable depending on what one values. Plurality's simplicity, for example, comes at the cost of not capturing backup preferences. That is a tradeoff. Vote splitting is not a feature anyone defends. It is a structural malfunction.

This distinction -- between pathologies that undermine the system's purpose and tradeoffs that reflect its priorities -- will recur throughout this series.

Plurality requires voters to concentrate support on a single candidate.

Section 5: Strategic Coordination

Because only one choice is allowed, plurality creates incentives for coordination.

Recall the earlier example:

If Progressive B's supporters had known their candidate would finish third, many might have supported Progressive A instead -- not because they preferred A, but because consolidating behind one progressive candidate was the only way to prevent a conservative victory.

This kind of reasoning -- choosing based on viability rather than genuine preference -- is called strategic voting.

Strategic voting occurs when a voter casts a ballot that does not reflect their genuine preferences because they believe a different ballot will produce a better outcome. It is not cheating. It is not a failure of the voter. It is rational behavior within the constraints of the system.

This dynamic is most visible in US presidential elections when third-party candidates enter the race. In 2000, voters who preferred Ralph Nader faced a question: vote for Nader, or vote for Al Gore to prevent a George W. Bush victory? In 1992, voters who preferred Ross Perot faced the same calculation in reverse. In each case, the choose-one ballot forced voters to weigh sincerity against strategy -- a decision imposed not by the candidates but by the structure of the ballot itself.

Voters may ask:

• Which candidate is viable?
• Should I support my favorite?
• Or should I support a compromise to avoid wasting my vote?

Every voting system creates some strategic landscape. The question is not whether strategy exists, but how much pressure the system places on voters to deviate from honest expression. Some systems -- as we will see later in this series -- are designed to reduce that pressure, making honest voting a stronger strategy than it is under plurality.

Political parties often emerge in part as coordination mechanisms under plurality systems.

Candidate Strategy

Voters are not the only actors who respond rationally to the structure of the ballot. Candidates face incentives shaped by the same rules.

Under plurality, winning requires only one thing: more votes than anyone else. A candidate does not benefit from being voters' second choice. The categorical ballot provides no mechanism for expressing that preference -- and so it counts for nothing.

This creates a direct incentive: concentrate support from a loyal base rather than seek broad acceptability.

A candidate who can mobilize 40 committed supporters will defeat a candidate who is acceptable to 60 voters but the first choice of only 35. Broad tolerability has no electoral value if tolerant voters have already marked someone else's name.

The rational response is base mobilization -- identifying and turning out the voters most likely to support you -- rather than seeking to minimize opposition. A candidate who carves out a distinct position and energizes that constituency is working with the system's logic. A candidate who tries to be broadly palatable may find that being acceptable is not enough when voters can only choose one name.

While we cannot attribute all candidate and campaign behavior to the voting system, we can identify the structural incentives it puts in place.

Section 6: The Two-Round Runoff

The problems described above -- non-majority winners, vote splitting, strategic coordination -- are not new observations.

One of the oldest structural responses is the two-round runoff.

How It Works

A two-round runoff adds a second election when no candidate wins a majority in the first round.

1️⃣ Round 1 -- All candidates compete. If one candidate receives a majority, they win outright.

2️⃣ Round 2 -- If no candidate achieves a majority, a second election is held. Typically, only the top two vote-getters from Round 1 advance. Voters return to choose between them.

Because the second round has only two candidates, the winner is guaranteed a majority.

A Familiar Structure

The two-round system is one of the most widely used alternatives to single-round plurality worldwide. France uses it for presidential and legislative elections. Many U.S. states and municipalities use a version of it, sometimes called a "runoff primary" or "general election runoff."

It is a direct attempt to combine plurality's simplicity (a categorical ballot, one choice per round) with a majority requirement.

What It Attempts to Solve

The two-round runoff addresses a central concern about plurality:

• In the first round, voters can express their genuine preference without strategic pressure, because a second round exists as a safeguard.
• In the second round, the field is narrowed to two candidates, ensuring majority support for the winner.

Vote splitting in the first round is less consequential, because the second round provides a corrective mechanism.

What It Introduces

The two-round runoff introduces tradeoffs of its own:

• A second election must be held. This increases administrative costs, extends the election timeline, and requires voters to participate twice.
• Turnout often drops in the second round. The majority winner of Round 2 may represent a majority of those who voted in the runoff -- but not necessarily a majority of those who voted in Round 1.
• The candidate field changes between rounds. Eliminated candidates may endorse a remaining candidate, shifting coalitions. Voters may reconsider their choices in light of new information or changed dynamics.
• First-round results still depend on vote splitting. If three similar candidates divide support in Round 1, all three may be eliminated -- and neither of the top-two finalists may represent that broader coalition.

The two-round runoff preserves the categorical ballot. Each round uses a simple choose-one format.

Its structural innovation is procedural: hold a second election to ensure majority support.

Its structural limitation is also procedural: that second election is costly, time-consuming, and not guaranteed to reflect the full electorate.

Section 7: What Plurality Optimizes For

Plurality voting prioritizes:

• Administrative simplicity
• Speed of tabulation
• Transparency of counting
• Low voter burden

It does not attempt to:

• Guarantee majority winners
• Capture backup preferences
• Measure intensity of support
• Resolve vote splitting

Its strength lies in procedural clarity.

Its tradeoffs appear when electorates fragment.

Section 8: Tradeoffs

Plurality voting:

• Is easy to understand
• Is easy to administer
• Produces decisive results in a single round

But it can:

• Elect candidates without majority support
• Penalize similar candidates
• Encourage strategic voting among voters
• Incentivize base mobilization over broad acceptability among candidates

These are structural consequences of the rule design.

Plurality balances simplicity against expressive capacity.

Conclusion

Plurality voting is the most widely used single-winner election method in the world.

It works cleanly in two-candidate contests and remains attractive because of its simplicity and transparency.

But when three or more candidates compete, plurality can produce outcomes in which:

• The winner lacks majority support.
• Similar candidates divide voters.
• Strategic coordination becomes central.

The two-round runoff is one of the oldest responses to these concerns. By adding a second election between the top two finishers, it guarantees a majority winner in the final round.

But a second election is costly. Turnout may decline. And the first round still relies on a categorical ballot, meaning vote splitting can still shape which candidates advance.

This raises a question:

What if the runoff could happen during counting -- without requiring voters to return for a second election?

In the next article, we examine one response:

Ranked Choice Voting.

Ranked Choice Voting (RCV)

Part A -- How It Works

Statement of Purpose

This article explains the mechanics of Ranked Choice Voting (RCV) for single-winner elections.

We will:

• Describe how ranked ballots are cast
• Walk through the counting process step by step
• Demonstrate a full multi-candidate example
• Introduce the locked ballot model

In the previous article, we examined Plurality Voting and the structural consequences that follow from its choose-one design.

Some reform efforts ask:

Can we move closer to majority outcomes without requiring a separate runoff election?

One response is Ranked Choice Voting.

Section 1: From Plurality to Ranked Choice Voting

Under plurality:

• Each voter selects one candidate.
• The candidate with the most votes wins.
• A majority is not required.

Plurality identifies the largest bloc of support in a single round.

But plurality does not attempt to:

• Guarantee majority winners
• Capture backup preferences
• Reduce vote splitting

Ranked Choice Voting adds a new dimension of information to the ballot: an ordered ranking of candidates.

Instead of selecting one name, voters indicate how candidates compare to one another.

What Is Ranked Choice Voting?

Ranked Choice Voting (RCV) allows voters to rank candidates in order of preference:

• 🥇 1st choice
• 🥈 2nd choice
• 🥉 3rd choice
• and so on.

For single-winner elections, the counting method used with ranked ballots is commonly called Instant Runoff Voting (IRV).

The Runoff Concept 🔁

A runoff election occurs when no candidate receives a majority in the first round of voting.

In traditional runoff systems:

1. Voters cast ballots.
2. If no candidate exceeds 50%, the top two candidates compete in a second election.
3. Voters return to vote again.

Ranked Choice Voting attempts to simulate this process within a single election.

➡️ The runoff happens during counting.

However:

In a traditional runoff, voters may reconsider their choices between rounds. In RCV, all rankings are fixed when the ballot is cast.

Section 2: How RCV Counting Works 🗳️

The counting process proceeds in rounds.

Step 1 -- Count First Choices 🥇

All first-choice votes are tallied.

If a candidate receives a majority of the ballots still in play (often called active ballots), they win.

At the beginning of the count, all ballots are still in play.

If no candidate has a majority, the process continues.

Step 2 -- Eliminate the Lowest Candidate ❌

The candidate with the fewest first-choice votes is eliminated.

Step 3 -- Transfer Ballots 🔄

Ballots that ranked the eliminated candidate first are transferred to the next ranked candidate who has not been eliminated.

Step 4 -- Repeat 🔁

After transfers:

• Totals are recalculated.
• If a candidate now has a majority, they win.
• If not, the next lowest candidate is eliminated.

Section 3: The Locked Ballot Model 🔒

At any given moment in the count:

A ballot counts for only one candidate.

Ballots remain locked onto their highest-ranked remaining candidate.

If that candidate is eliminated ❌:

• The ballot unlocks 🔓
• Moves to the next ranked candidate 🔄
• Locks again 🔒

Section 4: A 5-Candidate Example

A Note on Simplification

For clarity, this example groups voters into a small number of ranking patterns.

In a real five-candidate election, there are 120 different possible full ranking orders, and even more possible ballot types if voters rank only some candidates.

Real elections are usually much more varied than this simplified illustration -- but the counting process works the same way.

Candidates

• Alice 🤟🏼
• Ben 🫰🏼
• Carl ✌🏼
• Dale 👍🏼
• Erin 👌🏼

100 voters.

Ballots

• 30: Alice > Carl > Dale > Ben > Erin
• 24: Ben > Dale > Erin > Carl > Alice
• 20: Carl > Dale > Alice > Ben > Erin
• 16: Dale > Erin > Ben > Carl > Alice
• 10: Erin > Dale > Ben > Carl > Alice

Majority threshold: 51.

Round 1

Erin 👌🏼 is eliminated ❌.

Round 2

Erin's ballots transfer:

10: Erin ❌ → Dale 👍🏼

New totals:

Carl ✌🏼 is eliminated ❌.

Round 3

Carl's ballots transfer:

20: Carl ❌ → Dale 👍🏼

New totals:

Ben 🫰🏼 is eliminated ❌.

Round 4

Ben's ballots transfer:

24: Ben ❌ → Dale 👍🏼

Final totals:

Dale wins with a majority of active ballots.

What Just Happened?

Dale began in fourth place with only 16 first-choice votes.

Through four rounds of elimination and transfer, Dale accumulated support from voters whose earlier choices were eliminated.

No voter changed their ballot. The counting process revealed support that was already present but not visible in first-choice totals alone.

This is the core mechanical idea behind RCV: backup preferences become active as candidates are eliminated.

Conclusion

Ranked Choice Voting changes both the ballot and the counting process.

Instead of selecting one candidate, voters rank candidates in order of preference. Instead of a single tally, counting proceeds through rounds of elimination and transfer until one candidate holds a majority of active ballots.

The locked ballot model captures the key mechanic: each ballot counts for one candidate at a time, moving to the next ranked choice only when the current choice is eliminated.

These mechanics are straightforward.

But the design of the counting process -- sequential elimination based on first-choice totals -- produces structural consequences that are not always obvious from the rules alone.

In the next article, we examine those consequences:

What happens when a broadly acceptable candidate lacks first-choice support?

What happens when gaining additional support changes the elimination order?

And what happens when ballots run out of ranked candidates before the final round?

Ranked Choice Voting (RCV)

Part B -- Structural Consequences

Statement of Purpose

In the previous article, we examined how Ranked Choice Voting works: ranked ballots, sequential elimination, ballot transfers, and the locked ballot model.

The mechanics are straightforward.

But the design of the counting process -- sequential elimination based on first-choice totals -- produces structural consequences that are not always obvious from the rules alone.

This article examines three:

• A broadly acceptable candidate can be eliminated early.
• Gaining additional support can, in rare cases, change the outcome.
• Ballots can become inactive before the final round.

Understanding these consequences is essential for evaluating what RCV optimizes for -- and what it does not.

Section 1: The Center Squeeze 🎯

RCV eliminates candidates one at a time, starting with the fewest first-choice votes.

This means a candidate who is broadly acceptable -- but not many voters' first choice -- can be eliminated before their support becomes visible.

A Fully Worked Example

Three candidates:

• Lefty 🔴
• Center 🟡
• Righty 🔵

100 voters.

Ballots

• 40: Lefty > Center > Righty
• 20: Center > Lefty > Righty
• 40: Righty > Center > Lefty

Round 1

No candidate has a majority (51 required).

Center 🟡 has the fewest first-choice votes and is eliminated ❌.

Round 2

Center's 20 ballots transfer to their next ranked choice:

20: Center ❌ → Lefty 🔴

New totals:

Lefty wins with a majority.

But Consider the Head-to-Head Comparisons

What if the election had been only between Center and Lefty?

• 40 voters prefer Lefty over Center
• 20 voters prefer Center over Lefty
• 40 voters prefer Center over Lefty (Righty voters ranked Center second)

Center defeats Lefty, 60-40.

What about Center versus Righty?

• 40 voters prefer Lefty, who ranked Center second → prefer Center over Righty
• 20 voters prefer Center over Righty
• 40 voters prefer Righty over Center

Center defeats Righty, 60-40.

Center would have won against either opponent one-on-one.

Yet Center was eliminated first -- because RCV rewards concentrated first-choice support, not broad acceptability.

This is called the center squeeze: a broadly preferred candidate is squeezed out by opponents with stronger but narrower first-choice bases.

Section 2: When Increasing Support Changes Who Wins

In many voting systems, if additional voters rank a candidate higher -- and nothing else changes -- that candidate's position either improves or remains unchanged.

Voting theorists refer to this property as monotonicity.

A voting system is monotonic if, whenever a candidate would win under one set of ballots, that candidate would still win if some voters changed their ballots to rank that candidate higher, with no other changes.

Under this definition, increasing support cannot reverse a winning outcome.

Most single-round tally systems are monotonic.

Elimination-based systems operate differently.

Because candidates are removed one round at a time, increasing first-choice support can change which candidate is eliminated first. That shift can alter how later ballots are transferred.

In certain structured elections, a candidate who would have won may lose after gaining additional support.

A Fully Worked Example

Three candidates:

• Alice 🤟🏼
• Ben 🫰🏼
• Carl ✌🏼

100 voters.

Version 1 -- Original Election

• 36: Alice > Carl > Ben
• 35: Ben > Carl > Alice
• 29: Carl > Alice > Ben

Round 1

Carl is eliminated ❌.

Round 2

29: Carl ❌ → Alice 🤟🏼

Alice wins.

Version 2 -- Alice Gains Support

Now suppose 7 Ben voters switch to Alice.

• 43: Alice > Carl > Ben
• 28: Ben > Carl > Alice
• 29: Carl > Alice > Ben

Round 1

Ben is eliminated ❌.

Round 2

28: Ben ❌ → Carl ✌🏼

Carl wins.

What Happened?

Alice gained support -- but the elimination order changed -- and the outcome reversed.

In Version 1, Carl was eliminated first. In Version 2, Ben was eliminated first -- and Ben's voters transferred to Carl rather than Alice.

This requires a close race and similar levels of support among candidates. It does not occur in most elections, but it is a real structural possibility within elimination-based systems.

Section 3: When Ballots Leave the Count

Ranked Choice Voting does not require voters to rank all candidates.

When a ballot no longer lists any remaining candidates, it exhausts and can no longer influence subsequent rounds.

Active and Exhausted Ballots

• An active ballot lists at least one candidate who has not been eliminated.
• An exhausted ballot lists no remaining candidates.

Exhausted ballots were counted in earlier rounds. They simply no longer participate once all ranked candidates are eliminated.

The Two-Table Model 🗂️

Imagine the count taking place across two tables: one for ballots still in play, another for ballots that have exhausted.

Beginning of the Count

Majority threshold: 100 x 0.50 + 1 = 51

All 100 ballots are in play.

After Round 2

5 ballots exhaust.

New majority threshold: 95 x 0.50 + 1 = 48

After Round 3

10 additional ballots exhaust.

New majority threshold: 85 x 0.50 + 1 = 43

What Has Changed?

Originally, 51 votes were required -- a majority of 100 ballots cast.

After exhaustion, 43 votes are required -- a majority of 85 ballots still active.

The definition of majority has not changed.

What has changed is the number of ballots participating in the decisive round.

When we say that RCV produces a "majority winner," we mean a majority of the ballots still active in the final round.

Section 4: What Ranked Choice Voting Optimizes For

Ranked Choice Voting changes both the information collected on the ballot and the method used to process that information.

Compared to plurality, RCV:

• Allows voters to rank candidates rather than select only one.
• Simulates a runoff without requiring a second election.
• Transfers votes from eliminated candidates rather than splitting similar support.
• Produces winners who receive a majority of ballots still active in the final round.

Plurality identifies the largest bloc of support.

RCV attempts to consolidate support through successive rounds of elimination.

Structural Advantages

RCV can:

• Reduce vote splitting among similar candidates.
• Eliminate the need for a separate runoff election.
• Encourage voters to express backup preferences.
• Allow candidates to accumulate support beyond their initial base.

Structural Tradeoffs

Because RCV relies on sequential elimination:

• The order of elimination can affect the outcome.
• Broadly acceptable candidates may be eliminated early.
• Increasing support can, in rare cases, change who wins.
• Ballots may exhaust, reducing the number of ballots active in the final round.
• The majority threshold may differ from a majority of all ballots originally cast.

These are consequences of the rule structure.

Relationship to Majority and Consensus

RCV is designed to produce a majority winner under its counting framework.

Consensus, as examined in Part I, describes a condition of broad collective acceptance rather than a numerical threshold. RCV does not guarantee that condition.

RCV prioritizes majority consolidation through elimination rounds.

That design choice rearranges tradeoffs rather than eliminating them.

Conclusion

Ranked Choice Voting modifies the ballot and the counting rule in order to address vote splitting and produce majority outcomes without a second election.

It introduces structural consequences of its own:

• The center squeeze can eliminate broadly acceptable candidates.
• Gaining additional support can, in closely contested races, reverse outcomes.
• Ballot exhaustion can reduce the number of voters represented in the final round.

These are not errors in the system. They are products of its design.

Like plurality, RCV balances clarity, legitimacy, and expressiveness differently.

No voting system perfectly satisfies every desirable criterion. Each system defines what kind of agreement it is designed to measure.

But ranking is not the only way to express preference.

In the next article, we ask a broader question:

What if the ballot collected different information altogether -- not the order of preference, but the type and degree of support?

Approval Voting

A Different Way to Express Preference

Statement of Purpose

This article explains Approval Voting for single-winner elections. It describes how ballots are cast and counted, illustrates how the system addresses vote splitting, and evaluates the structural tradeoffs it introduces.

Approval Voting changes not how votes are counted -- but how voters express preference.

Section 1: Is Ranking the Only Way to Express Preference?

In the previous article, we examined Ranked Choice Voting (RCV), where voters rank candidates in order of preference.

Ranking allows voters to say:

• "I prefer A to B."
• "If A is eliminated, count me for C."

But ranking is not the only way to express support.

Another question is:

What if voters could support more than one candidate at the same time?

Instead of ordering candidates, voters could simply approve of any candidate they find acceptable.

That is the idea behind Approval Voting.

Section 2: How Approval Voting Works

Approval Voting ballots list all candidates.

Voters may:

• Approve one candidate
• Approve several candidates
• Approve all candidates
• Or approve only one

There is no ranking.

There is no scoring scale.

Each approved candidate receives one vote from that ballot.

The candidate with the most approvals wins.

No majority is required.

A Simple Example

Three candidates:

• Alice 🤟🏼
• Ben 🫰🏼
• Carl ✌🏼

100 voters.

Ballots:

40 voters approve: Alice 35 voters approve: Ben 25 voters approve: Carl

Alice wins.

So far, this looks identical to plurality voting.

The difference appears when voters approve more than one candidate.

Allowing Multiple Approvals

Suppose the same voters instead cast ballots this way:

40 voters approve: Alice and Carl 35 voters approve: Ben 25 voters approve: Carl

Now the totals are:

Carl wins.

Carl may not have been the first choice of most voters -- but many voters found Carl acceptable.

Approval Voting rewards broad acceptability.

Section 3: What Problem Does Approval Voting Attempt to Solve?

Vote Splitting

Under plurality voting, similar candidates can divide support.

Recall the earlier example:

The conservative wins even though 55 voters preferred a progressive candidate.

Under Approval Voting, progressive voters could approve both progressive candidates.

If those same voters approved both A and B:

Now one of the progressives wins (depending on tie-breaking rules).

Approval Voting reduces the spoiler effect because voters are not forced to choose only one acceptable option.

Section 4: Strategic Approval Thresholds

Approval Voting introduces a new decision for voters:

Where should I draw the line between "approve" and "do not approve"?

This is sometimes called an approval threshold.

Consider three candidates:

• Left 🔴
• Center 🟡
• Right 🔵

A voter who prefers Left most but finds Center acceptable must decide:

• Approve only Left?
• Approve Left and Center?

If they approve both, they increase Center's total -- possibly helping Center defeat Left.

If they approve only Left, they risk helping Right win.

Approval Voting reduces vote splitting -- but it introduces strategic judgment about how much compromise to signal.

Honesty and Strategy

In Part I, we defined strategic voting as casting a ballot that does not reflect genuine preferences in order to achieve a better outcome. Under plurality, strategy takes the form of abandoning a preferred candidate to consolidate support behind a viable one.

Approval Voting changes the shape of the strategic question. The decision is no longer "should I abandon my favorite?" but "should I extend support beyond my favorite?"

This distinction matters.

Under Approval Voting, a voter can always approve their genuine first choice. The strategic question is whether to also approve others. Approving your favorite is never a wasted vote -- a structural improvement over plurality, where supporting a non-viable candidate can directly help elect the candidate you like least.

Some systems are more strategy-resistant than others, meaning that honest expression is more frequently the optimal strategy. No system with three or more candidates is perfectly strategy-proof -- this is a proven mathematical result, much like Arrow's theorem. But "no system is perfectly strategy-proof" is very different from "all systems are equally strategic." Systems differ meaningfully in how much they reward deviation from honest expression.

This dimension -- how much a system incentivizes honest voting -- will continue to surface as we examine Score Voting and STAR Voting in later articles.

Section 5: Relationship to Majority Support

Approval Voting does not guarantee a majority winner.

A candidate can win with the highest number of approvals even if most voters preferred someone else as their top choice.

However, it often selects candidates with broad support because:

• Candidates benefit from being widely acceptable.
• Campaigns may seek second-tier approval.

Unlike Ranked Choice Voting:

• There is no elimination order.
• There are no transfers.
• There is no ballot exhaustion.

Every ballot counts equally in the final tally.

But voters must decide how much information to reveal.

Section 6: Tradeoffs

Approval Voting attempts to solve:

• Vote splitting
• The need for separate runoff elections
• The complexity of ranked ballots

It does so by:

• Allowing support for multiple candidates
• Counting approvals in a single round

But it introduces tradeoffs:

• Voters must choose an approval threshold.
• Strong compromise signaling may disadvantage a voter's favorite.
• It does not guarantee majority winners.
• It does not measure intensity of preference -- only acceptability.

Approval Voting is mechanically simple.

Its complexity lies in voter strategy.

Comparison to Ranked Choice Voting

RCV structures choice through elimination order.

Approval structures choice through simultaneous acceptability.

Each reduces vote splitting in different ways.

Each introduces distinct strategic considerations.

Conclusion

Approval Voting changes how voters express preference.

Instead of ranking candidates, voters indicate which candidates they find acceptable.

This can reduce vote splitting and reward broadly acceptable candidates.

At the same time, it shifts strategic decisions to the voter:

• How many candidates should I approve?
• Should I approve a compromise?

Approval Voting simplifies counting -- but it does not eliminate tradeoffs.

The next question is:

If we allow voters to express not just acceptability, but degrees of support -- what changes?

In the next article, we will examine Score Voting, which adds a rating scale to the ballot and introduces a new set of structural considerations.

Score Voting

Measuring Degrees of Support

Statement of Purpose

In the previous article, we examined Approval Voting, which allows voters to support more than one candidate.

Approval answers a simple question:

Which candidates are acceptable?

Score Voting asks a different one:

How strongly do you support each candidate?

This article explains how Score Voting works, demonstrates its counting process step by step, and evaluates the structural tradeoffs it introduces compared to Approval Voting and Ranked Choice Voting (RCV).

Section 1: What Score Voting Is

Score Voting (sometimes called Range Voting) uses a fixed numerical rating scale.

Common scales include:

• 0-5
• 0-10
• 0-100

The scale is defined before the election.

On the ballot, voters assign a score to each candidate. For example, on a 0-5 scale:

• 0 = lowest support
• 5 = highest support

Voters may:

• Give the same score to multiple candidates
• Give all candidates the same score
• Use only part of the scale

How Counting Works

1. Add all scores for each candidate.
2. (Sometimes totals are averaged, but totals and averages produce the same ranking.)
3. The candidate with the highest total score wins.

There are:

• No eliminations
• No ballot transfers
• No runoff stage
• A single round of tabulation

Every ballot contributes to every candidate's total.

Section 2: Fully Worked Example

Let's walk through a complete example.

Candidates:

• Alice 🤟🏼
• Ben 🫰🏼
• Carl ✌🏼

Voters: 100

Scale: 0-5

For clarity, we group voters by scoring pattern.

Ballot Groups

Group 1 -- 40 voters

Group 2 -- 35 voters

Group 3 -- 25 voters

Step 1 -- Multiply Scores by Voter Count

Alice

• 40 x 5 = 200
• 35 x 1 = 35
• 25 x 3 = 75

Total for Alice = 200 + 35 + 75 = 310

Ben

• 40 x 2 = 80
• 35 x 5 = 175
• 25 x 2 = 50

Total for Ben = 80 + 175 + 50 = 305

Carl

• 40 x 0 = 0
• 35 x 3 = 105
• 25 x 5 = 125

Total for Carl = 0 + 105 + 125 = 230

Final Totals

Alice wins with the highest aggregate score.

No candidates were eliminated. No ballots transferred. All 100 voters influenced the totals for all three candidates.

Section 3: What Problem Score Voting Attempts to Solve

Score Voting builds on concerns raised in earlier systems.

1️⃣ Vote Splitting

Like Approval Voting, Score Voting allows voters to support more than one candidate.

A voter who prefers Alice but finds Carl acceptable could score:

• Alice: 5
• Carl: 4

This reduces pressure to abandon acceptable alternatives in order to avoid splitting support.

2️⃣ Loss of Intensity

Approval Voting captures acceptability -- but not degree.

Ranking systems capture order -- but not strength.

Score Voting captures intensity of support.

3️⃣ Information Compression in Ranking

Ranking forces a strict order.

Score Voting allows ties and near-ties naturally through similar scores.

Section 4: Strategic Exaggeration

Because Score Voting uses a scale, voters must decide how to use it.

Honest scoring:

Possible exaggerated scoring:

When many voters compress the scale, Score Voting can resemble Approval Voting.

This is a structural incentive, not a counting error.

Section 5: Relationship to Majority Support

Score Voting does not require a majority threshold.

The candidate with the highest total score wins.

It optimizes for aggregate evaluation, not majority consolidation.

Section 6: Tradeoffs

Score Voting attempts to solve:

• Vote splitting
• Loss of intensity
• Binary approval limitation

It introduces:

• Strategic score exaggeration
• Scale interpretation variance
• No majority guarantee
• Less intuitive majority framing

Structural Comparison

Conclusion

Score Voting allows voters to express degrees of support.

It captures more information than Approval Voting.

It reduces vote splitting while introducing scale-based strategy considerations.

It is another branch in the voting system design space.

In the next article, we will examine STAR Voting, which combines scoring with an automatic runoff.

STAR Voting

A Hybrid of Scoring and Runoff Logic

Statement of Purpose

This article explains STAR Voting (Score Then Automatic Runoff) for single-winner elections.

STAR Voting combines two ideas we have already examined:

• Voters score candidates on a numerical scale.
• The top two scorers advance to an automatic runoff comparison.

We will:

• Walk through how ballots are cast and counted.
• Examine what problem STAR attempts to address.
• Compare it to Score Voting, Approval Voting, and Ranked Choice Voting.
• Evaluate the structural tradeoffs it introduces.

As in previous articles, the goal is clarity -- not advocacy.

Section 1: What Is STAR Voting?

STAR stands for Score Then Automatic Runoff.

Like Score Voting, voters rate each candidate on a fixed scale (commonly 0-5).

The counting happens in two stages:

Stage 1 -- Scoring Round

• Add up all scores for each candidate.
• The two candidates with the highest total scores advance.

Stage 2 -- Automatic Runoff

• For those two finalists only:
  • On each ballot, whichever finalist received the higher score is preferred.
• The finalist preferred on more ballots wins.

No separate election is held. The runoff is simulated using the same ballots.

Section 2: A Simple Example

Three candidates:

• Alice 🤟🏼
• Ben 🫰🏼
• Carl ✌🏼

100 voters. Scale: 0-5.

Ballots:

• 40 voters -- Alice: 5, Ben: 2, Carl: 0
• 35 voters -- Alice: 1, Ben: 5, Carl: 3
• 25 voters -- Alice: 3, Ben: 2, Carl: 5

Stage 1 -- Add Scores

Alice and Ben advance to the runoff.

Stage 2 -- Automatic Runoff

Now compare Alice and Ben ballot by ballot.

• 40 voters scored Alice (5) higher than Ben (2) → Alice preferred
• 35 voters scored Ben (5) higher than Alice (1) → Ben preferred
• 25 voters scored Alice (3) higher than Ben (2) → Alice preferred

Runoff totals:

Alice wins.

What Changed Compared to Score Voting?

In pure Score Voting, Alice already won by total score.

In STAR, that result is confirmed through a majority-style head-to-head comparison between the top two.

The second round ensures that the winner defeats the other finalist in a direct comparison.

Section 3: What Problem Does STAR Attempt to Solve?

STAR was developed in response to a concern about Score Voting.

The Concern: Strategic Exaggeration

In Score Voting, voters may exaggerate:

• Give favorite: 5
• Give competitor: 0

If many voters compress the scale this way, Score Voting begins to resemble Approval or Plurality Voting.

STAR attempts to reduce this incentive.

Because:

• Advancing to the top two depends on total score.
• Winning depends on being preferred in the runoff.

Giving a compromise candidate a moderate score can help them reach the runoff -- without necessarily causing them to defeat your favorite.

The two-stage structure aims to balance:

• Intensity (from scores)
• Broad pairwise support (from runoff comparison)

Section 4: Relationship to Majority Support

STAR does not require a majority in the scoring round.

But the final winner must defeat the other finalist in a head-to-head comparison of ballots.

In that sense, the winner:

• Has majority support against the other finalist,
• But not necessarily against every candidate in the field.

This differs from:

• Ranked Choice Voting, which produces a majority of active ballots after eliminations.
• Score Voting, which selects the highest total score without a runoff stage.

STAR adds a majority-style confirmation step -- but only between two candidates.

Section 5: Comparison to Other Systems

STAR combines:

• The expressive input of Score Voting
• A runoff-style majority check

It does not use elimination rounds. It does not transfer ballots. Every ballot counts in both stages.

Section 6: Structural Consequences

STAR addresses some concerns -- but introduces its own design effects.

1️⃣ Two Candidates Advance

Only the top two scorers enter the runoff.

A candidate who might defeat many opponents head-to-head may fail to advance if their total score is slightly lower.

2️⃣ Strategic Scoring Still Exists

Although STAR attempts to soften exaggeration incentives, voters still face decisions:

• Should I give a compromise a 4 or a 5?
• Could boosting a compromise push them into the runoff instead of my favorite?

Strategy is reduced in some contexts -- but not eliminated.

3️⃣ Majority Only Among Finalists

The winner defeats the other finalist -- but not necessarily all candidates.

This differs from systems designed to identify a candidate who would defeat every other candidate in one-on-one comparisons.

Section 7: Tradeoffs

STAR Voting attempts to balance:

• Expressiveness (via scoring)
• Simplicity of tabulation
• A majority-style final check

It avoids:

• Ballot exhaustion
• Sequential elimination order
• Separate runoff elections

But it introduces:

• A two-stage counting process
• Continued incentives around score strategy
• Dependence on which candidates reach the top two

Like all voting systems, STAR optimizes for certain goals:

• Strong overall support
• Direct comparison between leading candidates
• Single-election resolution

It does not eliminate tradeoffs. It rearranges them.

Conclusion

Plurality selects the most votes.

Ranked Choice Voting eliminates candidates until one has a majority of active ballots.

Approval Voting counts how many voters find each candidate acceptable.

Score Voting aggregates degrees of support.

STAR Voting combines scoring with an automatic runoff.

It seeks to preserve expressive ballots while ensuring the winner prevails in a head-to-head comparison between the top two scorers.

Whether this balance improves outcomes depends on:

• How voters use the scoring scale
• How competitive the field is
• What one values most in a voting system

In the final article of this series, we will examine a different lens altogether:

Rather than proposing a new ballot structure, we will explore a benchmark for evaluating systems -- the idea of a Condorcet winner.

Can a voting system guarantee that the candidate who would win every one-on-one contest is selected?

And what happens when preferences cycle?

Condorcet as Benchmark

A Standard for Evaluating Voting Systems

Statement of Purpose

Throughout this series, we examined several single-winner voting systems:

• Plurality
• Ranked Choice Voting (RCV)
• Approval Voting
• Score Voting
• STAR Voting

Each system altered either the ballot structure, the counting rule, or both.

This article introduces something different.

It does not propose a new ballot.

Instead, it introduces a theoretical benchmark -- a way to evaluate voting systems.

That principle is known as the Condorcet principle.

Section 1: Head-to-Head Majority Comparisons

To understand the Condorcet principle, we begin with a simple idea:

Compare candidates two at a time.

This is called a pairwise comparison.

Instead of asking "Who has the most first-choice votes?" or "Who has the highest score?" we ask:

If the election were only between Candidate A and Candidate B, who would win?

A Fully Worked Example

Three candidates:

• Alice 🤟🏼
• Ben 🫰🏼
• Carl ✌🏼

100 voters cast ranked ballots.

Ballot Groups

• 35 voters: Alice > Ben > Carl
• 33 voters: Ben > Carl > Alice
• 32 voters: Carl > Alice > Ben

No candidate has a majority of first-choice votes.

Instead of eliminating anyone, we calculate head-to-head totals.

Step 1 -- Alice vs Ben

Prefer Alice over Ben:

• 35 voters
• 32 voters

Total: 67

Prefer Ben over Alice:

• 33 voters

Result: Alice defeats Ben, 67-33.

Step 2 -- Alice vs Carl

Prefer Alice over Carl:

• 35 voters

Total: 35

Prefer Carl over Alice:

• 33 voters
• 32 voters

Total: 65

Result: Carl defeats Alice, 65-35.

Step 3 -- Ben vs Carl

Prefer Ben over Carl:

• 35 voters
• 33 voters

Total: 68

Prefer Carl over Ben:

• 32 voters

Result: Ben defeats Carl, 68-32.

Pairwise Matrix

No candidate defeats both opponents.

Section 2: The Condorcet Winner

A Condorcet winner is:

A candidate who would defeat every other candidate in one-on-one majority comparisons.

Not every election produces one.

When one exists, it represents a candidate preferred by a majority over each alternative.

Section 3: Cycles (The Condorcet Paradox)

Majority preferences can form a loop:

• A majority prefers Alice over Ben
• A majority prefers Ben over Carl
• A majority prefers Carl over Alice

This is called a Condorcet cycle.

It is a structural property of collective decision-making.

Section 4: Completion Methods

If a Condorcet winner exists, selecting them is straightforward.

If preferences cycle, additional rules -- called completion methods -- are required.

Different approaches resolve cycles differently. To see how, we can return to the cycle from Section 1 and apply one of the simplest completion methods.

Minimax: A Worked Example

One approach to resolving a cycle is called Minimax.

The idea: every candidate in a cycle loses at least one head-to-head matchup. Minimax selects the candidate whose worst loss is the smallest.

We already have the results from Section 1:

Now identify each candidate's worst defeat:

• Alice -- lost to Carl by 30 (65-35)
• Ben -- lost to Alice by 34 (67-33)
• Carl -- lost to Ben by 36 (68-32)

Alice's worst defeat is the smallest.

Under Minimax, Alice wins.

What This Illustrates

Minimax resolves the cycle by asking: which candidate has the strongest claim even in their weakest matchup?

Other completion methods use different logic. Some prioritize the strength of victories rather than the size of defeats. Some lock in the strongest pairwise results one at a time and discard weaker results that would create a cycle.

Each method produces a defensible winner -- but not always the same winner.

Even within the Condorcet family, design choices shape outcomes.

Section 5: Using Condorcet as a Criterion

A system passes the Condorcet criterion if it always elects the Condorcet winner when one exists.

Many commonly used systems do not guarantee this outcome.

The criterion is one evaluative dimension among many.

Section 6: Structural Comparison

Conclusion

The Condorcet principle provides a benchmark for evaluating voting systems.

It does not rank them.

It clarifies one structural question:

If a candidate would defeat every other candidate head-to-head, should the system select that candidate?

Voting systems reflect design priorities.

Criteria sometimes conflict. In 1951, the economist Kenneth Arrow proved that this is not merely an observation -- it is a mathematical certainty. No ranked voting system can simultaneously satisfy a small set of reasonable fairness criteria. This result, known as Arrow's Impossibility Theorem, is sometimes sensationalized as proof that "democracy is impossible." That framing misses the point. Just as no single food can satisfy all of a person's nutritional needs, no single voting method can satisfy everything we might want in a voting system. The impossibility is a design constraint, not a fatal flaw. It means that choosing a voting system requires understanding which criteria it prioritizes -- and which it does not.

Tradeoffs are unavoidable.

Understanding those tradeoffs is the purpose of this series.

Conclusion

From Single-Winner to Multi-Winner Systems

Statement of Purpose

This article concludes the single-winner portion of this series.

Across the previous articles, we examined multiple voting systems used to select one officeholder at a time. Each system changed either:

• The structure of the ballot
• The method of counting
• Or both

The goal was not to determine which system is "best."

It was to understand how design choices shape outcomes.

This final article synthesizes what we have learned -- and prepares us to expand the design space beyond single-winner elections.

What We Learned About Single-Winner Systems

Over the course of this series, we explored multiple structural dimensions.

Rather than summarizing by system, it is more useful to summarize by design choice.

1️⃣ Ballot Structure

We examined four distinct ways ballots can collect information:

Each ballot structure captures different information:

• Ranking captures order.
• Approval captures acceptability.
• Rating captures intensity.
• Choose-one captures only top preference.

More expressive ballots gather more information. They also change the strategic landscape -- not by introducing strategy where none existed, but by shifting the form that strategic reasoning takes.

2️⃣ Counting Logic

We examined several counting approaches:

• Plurality tally (single-round, most votes wins)
• Sequential elimination (RCV)
• Aggregate approval totals
• Aggregate scoring totals
• Score + automatic runoff (STAR)
• Pairwise head-to-head comparisons (Condorcet lens)

Each counting rule processes ballot information differently.

Some simulate runoffs. Some consolidate support. Some aggregate evaluation. Some check head-to-head viability.

The counting rule is not neutral. It determines what kind of support is decisive.

3️⃣ Majority vs Plurality

A central theme has been the difference between:

• Plurality winners (largest share)
• Majority winners (more than half)

Some systems guarantee a majority of active ballots. Others do not require one. Some confirm majority only between finalists. Some optimize for highest aggregate evaluation instead.

"Majority" itself turns out to be a defined threshold within a specific counting framework.

4️⃣ Expressiveness vs Simplicity

As ballots become more expressive, they ask more of voters:

• Rank fully?
• Approve strategically?
• Calibrate a score scale?

At the same time, more expressive systems often reduce pathologies -- systematic outcomes like vote splitting that no reasonable design would intentionally produce.

But they may introduce:

• Threshold decisions
• Score exaggeration incentives
• Elimination-order sensitivity

Simplicity and expressiveness are not opposites -- but they do trade off.

5️⃣ Strategic Incentives

No system eliminates strategy entirely.

Different systems relocate it:

• Plurality → coordination around viability
• RCV → ranking order and elimination effects
• Approval → approval threshold
• Score → scale compression
• STAR → score calibration + finalist positioning

Strategy changes form. It does not disappear.

6️⃣ Condorcet as Evaluative Lens

The final article introduced a benchmark:

If a candidate would defeat every other candidate in one-on-one majority comparisons, should that candidate win?

The Condorcet criterion provides an evaluative dimension -- not a complete design solution.

It reminds us that:

• Majority rule can be defined in different ways.
• Pairwise consistency is one structural value among many.
• Even majority-based systems must resolve cycles.

7️⃣ Voting Criteria: A Framework for Evaluation

The Condorcet criterion is one example of a broader concept: voting criteria.

A voting criterion is a specific, testable property that a voting system may or may not satisfy. Criteria provide a shared vocabulary for evaluating systems -- not by asking "which is best?" but by asking "what does this system guarantee, and what does it not?"

Throughout this series, we encountered several criteria by name or by demonstration, even when we did not always use formal labels.

Criteria Encountered in This Series

These four criteria appeared because they were directly relevant to the structural consequences we examined. But they are not the only criteria that voting theorists use.

Criteria Beyond This Series

The following criteria are well-established in voting theory but were outside the scope of a 101 introduction. They are listed here as a roadmap for further exploration.

No system satisfies all criteria simultaneously -- as Arrow's theorem confirms. But knowing which criteria a system satisfies, and which it sacrifices, is the foundation of informed evaluation.

This is the structural vocabulary that allows a reader to move beyond "which system is best?" and toward a more precise question: "best according to which priorities?"

Core Insight

No single-winner system optimizes every evaluative criterion simultaneously.

This is not just an empirical pattern. Arrow's Impossibility Theorem establishes it as a mathematical result: any voting system that uses ranked preferences must sacrifice at least one of a set of basic fairness properties. The question is not whether tradeoffs exist, but which ones a given system accepts.

Different systems prioritize:

• Concentrated first-choice support
• Majority consolidation
• Broad acceptability
• Intensity of support
• Pairwise dominance

When one value is strengthened, another may weaken.

Tradeoffs of this kind are inherent to institutional design. The central task is recognizing which priorities a system elevates -- and which it does not.

The Limits of Single-Winner Design

All systems we examined share a defining feature:

They select one officeholder.

This constraint shapes every tradeoff.

When a group must choose a single individual:

• Preferences are compressed.
• Coalitions consolidate.
• Minor factions may go unrepresented.
• Compromise is forced into a single outcome.

Some tensions we observed -- such as vote splitting, elimination order sensitivity, or threshold strategy -- arise within this structural constraint.

But other tensions stem from the constraint itself.

When only one position exists:

• Representation is zero-sum.
• One coalition governs.
• Others do not.

Single-winner systems differ in how they determine who that coalition will be.

They do not change the fact that only one coalition ultimately governs.

Recognizing this helps clarify a broader point:

Ballot format is only one dimension of institutional design.

The number of seats available is another.

Preview: Multi-Winner Methods

So far, the design problem has been:

How should we select one winner?

Multi-winner systems change the question:

How should representation be allocated across multiple seats?

This shift introduces new structural concepts.

District Magnitude

District magnitude refers to how many representatives are elected from a district.

• Magnitude 1 → single-winner
• Magnitude 3, 5, 10, etc. → multi-winner

As magnitude increases, the design problem changes.

Majority vs Proportional Logic

Single-winner systems often prioritize:

• Identifying a majority-supported individual
• Or selecting the most broadly supported candidate

Multi-winner systems often introduce:

• Proportional representation
• Allocation of seats to reflect group support

Instead of compressing preferences into one outcome, multi-winner systems may distribute representation across factions.

Coalition Representation

In multi-winner systems:

• Groups may gain seats roughly proportional to their support.
• Representation can be shared rather than concentrated.
• Majority and minority blocs may both hold seats.

This does not eliminate tradeoffs.

It changes the type of tradeoffs being considered.

The design objective shifts from:

Who wins?

To:

How should representation be distributed?

The Broader Design Space

Voting systems are part of institutional architecture.

They interact with:

• District size
• Number of seats
• Party systems
• Legislative rules
• Executive structure

Changing ballot format alters incentives.

Changing district magnitude alters representation itself.

Tradeoffs do not disappear in multi-winner systems.

They reappear in new forms:

• Simplicity vs proportionality
• Local accountability vs coalition diversity
• Stability vs fragmentation

Criteria may conflict differently at larger scales.

But they still conflict.

Expanding the Frame

Throughout this series, one theme has remained constant:

Voting systems reflect priority choices.

Each design:

• Collects certain information
• Processes it in a particular way
• Guarantees some outcomes
• Leaves others unguaranteed

Evaluative criteria sometimes conflict:

• Majority consolidation
• Broad acceptability
• Intensity of support
• Pairwise consistency
• Simplicity
• Proportional fairness

No single-winner system satisfies all simultaneously.

Understanding this is structural literacy.

As we move forward, we expand the design space.

The question is no longer only:

Which individual should hold office?

It becomes:

How should representation itself be structured?

Multi-winner systems do not solve the tradeoffs we have studied.

They reconfigure them.

The next phase of this series will examine how expanding the number of seats changes the nature of democratic choice -- and what new design questions emerge when representation is shared rather than singular.